A074715 Number of prime factors of F(2^n) where F(m) is the m-th Fibonacci number.

0, 1, 2, 3, 4, 6, 8, 10, 13, 17, 20

Offset: 1

Author

Benoit Cloitre, Sep 04 2002

Keywords

• nonn
• more
• hard

Comments

a(11) = 20: F(2^11)/F(2^10) = 10077 286735 077005 660982 008061 065073 068074 475300 466012 444629 388487 574769 652115 651763 500026 128367 679301 744790 365920 278775 601766 000217 455997 930809 875108 639504 578766 853603 625505 162682 177708 433023 235042 368022 152858 871807 = 85 386449 571091 580927 (Curve 32) x 2359 309429 082740 633601 (Curve 34) x 50023 002657 441668 505734 051281 798632 006164 815525 563108 785740 688607 215220 816049 385513 420452 786904 754250 297381 898342 562475 311659 851996 080980 941281 231807 930745 342088 837303 971841

From Robert Israel, Apr 15 2015: (Start)

a(n+1) > a(n) since gcd(F(2^(n+1)), F(2^n)) = F(2^n).

a(n+1) = a(n) + 1 iff F(2^(n+1))/F(2^n) = F(2^n-1) + F(2^n+1) is prime, which is true for n <= 4 but not from n = 5 to at least 17. (End)

Prime factors counted with multiplicity. - Harvey P. Dale, Jun 09 2024

Examples

			a(10) = 17: 4506 699633 677819 813104 383235 728886 049367 860596 218604 830803 023149 600030 645708 721396 248792 609141 030396 244873 266580 345011 219530 209367 425581 019871 067646 094200 262285 202346 655868 899711 089246 778413 354004 103631 553925 405243 = 3 x 7 x 47 x 127 x 1087 x 2207 x 4481 x 21503 x 34303 x 119809 x 73327 699969 (Curve 2) x 186812 208641 x 455666 699738 584063 (Curve 27) x 4698 167634 523379 875583 x 1 019850 606646 830767 915009 (Curve 318) x 125 960894 984050 328038 716298 487435 392001 x 10 045901 211945 615185 770822 340063 765796 721091 525356 133475 714047
		

Crossrefs

Cf. A001221 (omega(n)), A058635 (Fibonacci(2^n)).

Programs

Mathematica

PrimeOmega[Fibonacci[2^Range[11]]] (* Harvey P. Dale, Jun 09 2024 *)

PARI

a(n)=omega(fibonacci(2^n))

Formula

a(n) = A001221(A058635(n)). - Michel Marcus, Apr 15 2015

Extensions

a(10) and a(11) from Jorge Coveiro, Jan 28 2006

Entry revised by N. J. A. Sloane, Feb 17 2006

a(9) corrected by Kellen Shenton, May 20 2022